Pascal's Wager

First published Sat May 2, 1998; substantive revision Fri Sep 1, 2017

“Pascal’s Wager” is the name given to an argument
due to Blaise Pascal for believing, or for at least taking steps to
believe, in God. The name is somewhat misleading, for in a single
section of his Pensées, Pascal apparently presents at
least three such arguments, each of which might be called a
‘wager’—it is only the final of these that is
traditionally referred to as “Pascal’s Wager”. We
find in it the extraordinary confluence of several important strands
of thought: the justification of theism; probability theory and
decision theory, used here for almost the first time in history;
pragmatism; voluntarism (the thesis that belief is a matter of the
will); and the use of the concept of infinity.

We will begin with some brief stage-setting: some historical
background, some of the basics of decision theory, and some of the
exegetical problems that the Pensées pose. Then we will
follow the text to extract three main arguments. The bulk of the
literature addresses the third of these arguments, as will the bulk of
our discussion here. Some of the more technical and scholarly aspects
of our discussion will be relegated to lengthy footnotes, to which
there are links for the interested reader. All quotations are from
§233 of Pensées (1910, Trotter translation), the
‘thought’ whose heading is
“Infinite—nothing”.

1. Background

It is important to contrast Pascal’s argument with various putative
‘proofs’ of the existence of God that had come before it.
Anselm’s ontological argument, Aquinas’ ‘five ways’,
Descartes’ ontological and cosmological arguments, and so on, purport
to prove that God exists. Pascal is
apparently unimpressed by such attempted justifications of theism:
“Endeavour … to convince yourself, not by increase of proofs of
God…” Indeed, he concedes that “we do not know if He is …”.
Pascal’s project, then, is radically different: he seeks to provide
prudential reasons for believing in God. To put it simply, we
should wager that God exists because it is the best bet. Ryan
1994 finds precursors to this line of reasoning in the writings of
Plato, Arnobius, Lactantius, and others; we might add Ghazali to his
list—see Palacios 1920. But what is distinctive is Pascal’s
explicitly decision-theoretic formulation of the reasoning. In fact,
Hacking 1975 describes the Wager as “the first well-understood
contribution to decision theory” (viii). Thus, we should pause briefly
to review some of the basics of that theory.

In any decision problem, the way the world is, and what an agent
does, together determine an outcome for the agent. We may assign
utilities to such outcomes, numbers that represent the degree
to which the agent values them. It is typical to present these numbers
in a decision matrix, with the columns corresponding to the various
relevant states of the world, and the rows corresponding to the various
possible actions that the agent can perform.

In decisions under uncertainty, nothing more is
given—in particular, the agent does not assign subjective
probabilities to the states of the world. Still, sometimes rationality
dictates a unique decision nonetheless. Consider, for example, a case
that will be particularly relevant here. Suppose that you have two
possible actions, \(A_1\) and \(A_2\), and the worst outcome
associated with \(A_1\) is at least as good as the best outcome associated
with \(A_2\); suppose also that in at least one state of the world,
\(A_1\)’s outcome is strictly better than \(A_2\)’s. Let us say in
that case that \(A_1\)
superdominates \(A_2\). Then rationality seems to require you to
perform \(A_1\).[1]

In decisions under risk, the agent assigns subjective
probabilities to the various states of the world. Assume that the
states of the world are independent of what the agent does. A figure of
merit called the expected utility, or the expectation
of a given action can be calculated by a simple formula: for each
state, multiply the utility that the action produces in that state by
the state’s probability; then, add these numbers. According to decision
theory, rationality requires you to perform the action of maximum
expected utility (if there is one).

Example. Suppose that the utility of money is
linear in number of dollars: you value money at exactly its face value.
Suppose that you have the option of paying a dollar to play a game in
which there is an equal chance of returning nothing, and returning
three dollars. The expectation of the game itself is

\[
0 \times \frac{1}{2} + 3 \times \frac{1}{2} = 1.5,
\]

so the expectation of paying a dollar for certain, then playing, is

\[
-1 + 1.5 = 0.5
\]

This exceeds the expectation of not playing (namely 0), so you should
play. On the other hand, if the game gave an equal chance of returning
nothing, and returning two dollars, then its expectation would be:

\[
0 \times \frac{1}{2} + 2 \times \frac{1}{2} = 1.
\]

Then consistent with decision theory, you could either pay the dollar
to play, or refuse to play, for either way your overall expectation
would be 0.

Considerations such as these will play a crucial role in Pascal’s
arguments. It should be admitted that there are certain exegetical
problems in presenting these arguments. Pascal never finished the
Pensées, but rather left them in the form of notes of
various sizes pinned together. Hacking 1972 describes the
“Infinite—nothing” as consisting of “two pieces of paper covered
on both sides by handwriting going in all directions, full of erasures,
corrections, insertions, and afterthoughts”
(24).[2]
This may explain why certain passages are notoriously difficult to
interpret, as we will see. Furthermore, our formulation of the
arguments in the parlance of modern Bayesian decision theory might
appear somewhat anachronistic. For example, Pascal did not
distinguish between what we would now call objective and
subjective probability, although it is clear that it is the
latter that is relevant to his arguments. To some extent, “Pascal’s
Wager” now has a life of its own, and our presentation of it here is
perfectly standard. Still, we will closely follow Pascal’s text,
supporting our reading of his arguments as much as possible. (See also Golding 1994 for another detailed analysis of Pascal’s reasoning, broken down into more steps than the presentation here.)

There is the further problem of dividing the
Infinite-nothing into separate arguments. We will locate three
arguments that each conclude that rationality requires you to wager for
God, although they interleave in the
text.[3]
Finally, there is some disagreement over just what “wagering for God”
involves—is it believing in God, or merely
engendering belief? We will conclude with a discussion of what Pascal
meant by this.

2. The Argument from Superdominance

Pascal maintains that we are incapable of knowing whether God exists or
not, yet we must “wager” one way or the other. Reason cannot settle
which way we should incline, but a consideration of the relevant
outcomes supposedly can. Here is the first key passage:

“God is, or He is not.” But to which side shall we incline?
Reason can decide nothing here. There is an infinite chaos which
separated us. A game is being played at the extremity of this infinite
distance where heads or tails will turn up… Which will you choose
then? Let us see. Since you must choose, let us see which interests you
least. You have two things to lose, the true and the good; and two
things to stake, your reason and your will, your knowledge and your
happiness; and your nature has two things to shun, error and misery.
Your reason is no more shocked in choosing one rather than the other,
since you must of necessity choose… But your happiness? Let us weigh
the gain and the loss in wagering that God is… If you gain, you gain
all; if you lose, you lose nothing. Wager, then, without hesitation
that He is.

There are exegetical problems already here, partly because Pascal
appears to contradict himself. He speaks of “the true” as
something that you can “lose”, and “error” as
something “to shun”. Yet he goes on to claim that if you
lose the wager that God is, then “you lose
nothing”. Surely in that case you “lose the true”,
which is just to say that you have made an error. Pascal believes, of
course, that the existence of God is “the
true”—but that is not something that he can appeal to in this
argument. Moreover, it is not because “you must of necessity
choose” that “your reason is no more shocked in choosing
one rather than the other”. Rather, by Pascal’s own account, it
is because “[r]eason can decide nothing here”. (If it
could, then it might well be shocked—namely, if you chose in a
way contrary to it.)

Following McClennen 1994, Pascal’s argument seems to be best
captured as presenting the following decision matrix:

God exists

God does not exist

Wager for God

Gain all

Status quo

Wager against God

Misery

Status quo

Wagering for God superdominates wagering against God: the worst
outcome associated with wagering for God (status quo) is at least as
good as the best outcome associated with wagering against God (status
quo); and if God exists, the result of wagering for God is strictly
better than the result of wagering against God. (The fact that the
result is much better does not matter yet.) Pascal draws the
conclusion at this point that you should wager for
God.

Without any assumption about your probability assignment to God’s
existence, the argument is invalid. Rationality does not
require you to wager for God if you assign probability 0 to God
existing, as a strict atheist might. And Pascal does not explicitly rule this possibility out
until a later passage, when he assumes that you assign positive
probability to God’s existence; yet this argument is presented as if it
is self-contained. His claim that “[r]eason can decide nothing here”
may suggest that Pascal regards this as a decision under uncertainty,
which is to assume that you do not assign probability at all
to God’s existence. If that is a further premise, then the argument is apparently
valid; but that premise contradicts his subsequent assumption that you
assign positive probability. See McClennen for a reading of this
argument as a decision under uncertainty.

Pascal appears to be aware of a further objection to this argument,
for he immediately imagines an opponent replying:

“That is very fine. Yes, I must wager; but I may perhaps
wager too much.”

The thought seems to be that if I wager for God, and God does not
exist, then I really do lose something. In fact, Pascal himself speaks
of stakingsomething when one wagers for God, which
presumably one loses if God does not exist. (We have already mentioned
‘the true’ as one such thing; Pascal also seems to regard
one’s worldly life as another.) In that case, the matrix is mistaken
in presenting the two outcomes under ‘God does not exist’
as if they were the same, and we do not have a case of superdominance
after all.

Pascal addresses this at once in his second argument, which we will
discuss only briefly, as it can be thought of as just a prelude to the
main argument.

3. The Argument From Expectation

He continues:

Let us see. Since there is an equal risk of gain and of
loss, if you had only to gain two lives, instead of one, you might
still wager. But if there were three lives to gain, you would have to
play (since you are under the necessity of playing), and you would be
imprudent, when you are forced to play, not to chance your life to gain
three at a game where there is an equal risk of loss and gain. But
there is an eternity of life and happiness.

His hypothetically speaking of “two lives” and
“three lives” may strike one as odd. It is helpful to bear
in mind Pascal’s interest in gambling (which after all provided the
initial motivation for his study of probability) and to take the
gambling model quite seriously here. Indeed, the Wager is permeated
with gambling metaphors: “game”, “stake”,
“heads or tails”, “cards” and, of course,
“wager”. Now, recall our calculation of the expectations
of the two dollar and three dollar gambles. Pascal apparently assumes
now that utility is linear in number of lives, that wagering
for God costs “one life”, and then reasons analogously to
the way we did in our expectation calculations above! This is, as it were, a warm-up. Since wagering for
God is rationally required even in the hypothetical case in which one
of the prizes is three lives, then all the more it is rationally
required in the actual case, in which one of the prizes is
an eternity of life (salvation).

So Pascal has now made two striking assumptions:

The probability of God’s existence is 1/2.

Wagering for God brings infinite reward if God
exists.

Morris 1994 is sympathetic to (1), while Hacking 1972 finds it “a
monstrous premiss”. One way to defend it is via the classical
interpretation of probability, according to which all possibilities are
given equal weight. The interpretation seems attractive for various gambling games, which by design involve an evidential symmetry with respect to their outcomes; and Pascal even likens God’s existence to a coin toss, evidentially speaking. However, unless more is said, the interpretation
yields implausible, and even contradictory results. (You have a
one-in-a-million chance of winning the lottery; but either you win the
lottery or you don’t, so each of these possibilities has probability
1/2?!) Pascal’s argument for (1) is presumably that “[r]eason can
decide nothing here”. (In the lottery ticket case, reason can decide
something.) But it is not clear that complete ignorance
should be modeled as sharp indifference. Morris imagines, rather, an
agent who does have evidence for and against the existence of God, but
it is equally balanced. In any case, it \(is\) clear that there
are people in Pascal’s audience who do not assign probability 1/2 to
God’s existence. This argument, then, does not speak to them.

However, Pascal realizes that the value of 1/2 actually plays no
real role in the argument, thanks to (2). This brings us to the third,
and by far the most important, of his arguments.

4. The Argument From Generalized Expectations: “Pascal’s Wager”

We continue the quotation.

But there is an eternity of life and happiness. And this
being so, if there were an infinity of chances, of which one only would
be for you, you would still be right in wagering one to win two, and
you would act stupidly, being obliged to play, by refusing to stake one
life against three at a game in which out of an infinity of chances
there is one for you, if there were an infinity of an infinitely happy
life to gain. But there is here an infinity of an infinitely happy life
to gain, a chance of gain against a finite number of chances of loss,
and what you stake is finite. It is all divided; wherever the infinite
is and there is not an infinity of chances of loss against that of
gain, there is no time to hesitate, you must give all…

Again this passage is difficult to understand completely. Pascal’s talk
of winning two, or three, lives is a little misleading. By his own
decision theoretic lights, you would not act stupidly “by
refusing to stake one life against three at a game in which out of an
infinity of chances there is one for you”—in fact, you should not
stake more than an infinitesimal amount in that case (an amount that is
bigger than 0, but smaller than every positive real number). The point,
rather, is that the prospective prize is “an infinity of an infinitely
happy life”. In short, if God exists, then wagering for God results in
infinite utility.

What about the utilities for the other possible outcomes? There is
some dispute over the utility of “misery”. Hacking interprets this as
“damnation”, and Pascal does later speak of “hell” as the outcome in
this case. Martin 1983 among others assigns this a value of
negative infinity. Sobel 1996, on the other hand, is one
author who takes this value to be finite. There is some textual support
for this reading: “The justice of God must be vast like His compassion.
Now justice to the outcast is less vast … than mercy towards the
elect”. As for the utilities of the outcomes associated with God’s
non-existence, Pascal tells us that “what you stake is finite”. This
suggests that whatever these values are, they are finite.

Pascal’s guiding insight is that the argument from expectation goes
through equally well whatever your probability for God’s
existence is, provided that it is non-zero and finite
(non-infinitesimal)—“a chance of gain against a finite number
of chances of
loss”.[4]

Pascal’s assumptions about utilities and probabilities are now in
place. In another landmark moment in this passage, he next presents a
formulation of expected utility theory. When gambling, “every
player stakes a certainty to gain an uncertainty, and yet he stakes a
finite certainty to gain a finite uncertainty, without transgressing
against reason”. How much, then, should a player be prepared to
stake without transgressing against reason? Here is Pascal’s answer:
“… the uncertainty of the gain is proportioned to the
certainty of the stake according to the proportion of the chances of
gain and loss …” It takes some work to show that this
yields expected utility theory’s answer exactly, but it is work well
worth doing for its historical
importance.[5]
(The interested reader can see this work done at
footnote 5.)

Let us now gather together all of these points into a single
argument. We can think of Pascal’s Wager as having three premises: the
first concerns the decision matrix of rewards, the second concerns the
probability that you should give to God’s existence, and the third is a
maxim about rational decision-making. Specifically:

Either God exists or God does not exist, and you can either wager
for God or wager against God. The utilities of the relevant possible
outcomes are as follows, where \(f_1, f_2\), and
\(f_3\) are numbers whose values are not specified beyond the
requirement that they be finite:

God exists

God does not exist

Wager for God

\(\infty\)

\(f_1\)

Wager against God

\(f_2\)

\(f_3\)

Rationality requires the probability that you assign to God
existing to be positive, and not infinitesimal.

Rationality requires you to perform the act of maximum expected
utility (when there is one).

Conclusion 1. Rationality requires you to wager for
God.

Conclusion 2. You should wager for God.

We have a decision under risk, with probabilities assigned to the ways the world could be, and utilities assigned to the outcomes. In particular, we represent the infinite utility associated with salvation as ‘\(\infty\)’. We assume that the real line is extended to include the element ‘\(\infty\)’, and that the basic arithmetical operations are extended as follows:

This is
finite.[6]
By premise 3, rationality requires
you to perform the act of maximum expected utility. Therefore,
rationality requires you to wager for God.

We now survey some of the main objections to the argument.

5. Objections to Pascal’s Wager

5.1 Premise 1: The Decision Matrix

Here the objections are manifold. Most of them can be stated
quickly, but we will give special attention to what has generally been
regarded as the most important of them, ‘the many Gods
objection’ (see also the link to footnote 7).

1. Different matrices for different people. The argument
assumes that the same decision matrix applies to everybody. However,
perhaps the relevant rewards are different for different people.
Perhaps, for example, there is a predestined infinite reward for the
Chosen, whatever they do, and finite utility for the rest, as Mackie
1982 suggests. Or maybe the prospect of salvation appeals more to some
people than to others, as Swinburne 1969 has noted.

Even granting that a single \(2 \times 2\) matrix applies to
everybody, one might dispute the values that enter into it. This
brings us to the next two objections.

2. The utility of salvation could not be infinite. One might
argue that the very notion of infinite utility is suspect—see
for example Jeffrey 1983 and McClennen
1994.[7]
Hence, the
objection continues, whatever the utility of salvation might be, it
must be finite. Strict finitists, who are suspicious of the notion of
infinity in general, will agree—see Dummett 1978 and Wright
1987. Or perhaps the notion of infinite utility makes sense, but an
infinite reward could only be finitely appreciated by a human
being.

3. There should be more than one infinity in the matrix. There
are also critics of the Wager who, far from objecting to infinite
utilities, want to see more of them in the matrix. For
example, it might be thought that a forgiving God would bestow
infinite utility upon wagerers-for and wagerers-against
alike—Rescher 1985 is one author who entertains this
possibility. Or it might be thought that, on the contrary, wagering
against an existent God results in negative infinite
utility. (As we have noted, some authors read Pascal himself as saying
as much.) Either way, \(f_2\) is not really finite at all, but
\(\infty\) or \(-\infty\) as the case may be. And perhaps \(f_1\) and
\(f_3\) could be \(\infty\) or \(-\infty\). Suppose, for instance,
that God does not exist, but that we are reincarnated ad
infinitum, and that the total utility we receive is an infinite
sum that diverges to infinity or to negative infinity.

4. The matrix should have more rows. Perhaps there is more
than one way to wager for God, and the rewards that God bestows vary
accordingly. For instance, God might not reward infinitely those who
strive to believe in Him only for the very mercenary reasons that
Pascal gives, as James 1956 has observed. One could also imagine
distinguishing belief based on faith from belief based on evidential
reasons, and posit different rewards in each case.

5. The matrix should have more columns: the many Gods
objection. If Pascal is really right that reason can decide nothing
here, then it would seem that various other theistic hypotheses are
also live options. Pascal presumably had in mind the Catholic
conception of God—let us suppose that this is the God who
either ‘exists’ or ‘does not exist’. By
excluded middle, this is a partition. The objection, then, is that the
partition is not sufficiently fine-grained, and the ‘(Catholic)
God does not exist’ column really subdivides into various
other theistic hypotheses. The objection could equally run
that Pascal’s argument ‘proves too much’: by parallel
reasoning we can ‘show’ that rationality requires believing
in various incompatible theistic hypotheses. As Diderot (1746) puts
the point: “An Imam could reason just as well this
way”.[8]

Since then, the point has been presented again and refined in various
ways. Mackie 1982 writes, “the church within which alone salvation is
to be found is not necessarily the Church of Rome, but perhaps that of
the Anabaptists or the Mormons or the Muslim Sunnis or the worshippers
of Kali or of Odin” (203). Cargile 1966 shows just how easy it is to
multiply theistic hypotheses: for each real number \(x\), consider
the God who prefers contemplating \(x\) more than any other
activity. It seems, then, that such ‘alternative gods’ are
a dime a dozen—or \(\aleph_1\), for that matter.

In response, some authors argue that in such a competition among
various possible deities for one’s belief, some are more probable than
others. Although there may be ties among the expected
utilities—all infinite—for believing in various ones among them,
their respective probabilities can be used as
tie-breakers. Schlesinger (1994, 90) offers this principle: “In
cases where the mathematical expectations are infinite, the criterion for
choosing the outcome to bet on is its probability”. (Note that this
principle is not found in the Wager itself, although it might be regarded as
a friendly addition.) Are there
reasons, then, for assigning higher probability to some Gods than
others? Jordan (1994a, 107) suggests that some outlandish theistic
hypotheses may be dismissed for having “no backing of
tradition”. Similarly, Schlesinger maintains that Pascal is
addressing readers who “have a notion of what genuine religion
is about” (88), and we might take that to suggest that Cargile’s
imagined Gods, for example, may be correspondingly assigned lower
probability than Pascal’s God. Lycan and Schlesinger 1989 give more
theoretical reasons for favoring Pascal’s God over others in one’s
probability assignments. They begin by noting the familiar problem in
science of underdetermination of theory by evidence. Faced with a
multiplicity of theories that all fit the observed data equally well,
we favor the simplest such theory. They go on to argue that simplicity
considerations similarly favor a conception of God as
“absolutely perfect”, “which is theologically unique
in that it implies all the other predicates traditionally ascribed to
God” (104), and we may add that this conception is
Pascal’s. Conceptions of rival Gods, by contrast, leave open various
questions about their nature, the answering of which would detract
from their simplicity, and thus their probability.

Finally, Bartha 2012 models one’s probability assignments to
various theistic hypotheses as evolving over time according to a
‘deliberational dynamics’ somewhat analogous to the
dynamics of evolution by natural selection. So understood, Pascal’s
Wager is not a single decision, but rather a sequence of decisions in
which one’s probabilities update sequentially in proportion to how
choiceworthy each God appeared to be in the previous round. (This
relies on a sophisticated handling of infinite utilities in terms of
utility ratios given in his 2007; see below.) He argues that a given
probability assignment is choiceworthy only if it is an equilibrium of
this deliberational dynamics. He shows that certain assignments are
choiceworthy by this criterion, thus providing a kind of vindication
of Pascal against the many Gods objection.

5.2 Premise 2: The Probability Assigned to God’s Existence

There are four sorts of problem for this premise. The first two are
straightforward; the second two are more technical, and can be found by
following the link to footnote 9.

1. Undefined probability for God’s existence. Premise 1
presupposes that you should have a probability for God’s
existence in the first place. However, perhaps you could rationally
fail to assign it a probability—your probability that
God exists could remain undefined. We cannot enter here into
the thorny issues concerning the attribution of probabilities to
agents. But there is some support for this response even in Pascal’s
own text, again at the pivotal claim that “[r]eason can decide nothing
here. There is an infinite chaos which separated us. A game is being
played at the extremity of this infinite distance where heads or tails
will turn up…” The thought could be that any probability assignment
is inconsistent with a state of “epistemic nullity” (in Morris’ 1986
phrase): to assign a probability at all—even 1/2—to
God’s existence is to feign having evidence that one in fact totally
lacks. For unlike a coin that we know to be fair, this metaphorical
‘coin’ is ‘infinitely far’ from us, hence
apparently completely unknown to us. Perhaps, then, rationality
actually requires us to refrain from assigning a probability
to God’s existence (in which case at least the Argument from
Superdominance would apparently be valid). Or perhaps rationality does not require
it, but at least permits it. Either way, the Wager would not
even get off the ground.

2. Zero probability for God’s existence. Strict atheists may
insist on the rationality of a probability assignment of 0, as Oppy
1990 among others points out. For example, they may contend that reason
alone can settle that God does not exist, perhaps by arguing
that the very notion of an omniscient, omnipotent, omnibenevolent being
is contradictory. Or a Bayesian might hold that rationality places no
constraint on probabilistic judgments beyond coherence (or conformity
to the probability calculus). Then as long as the strict atheist
assigns probability 1 to God’s non-existence alongside his or her
assignment of 0 to God’s existence, no norm of rationality has been
violated.

Furthermore, an assignment of \(p = 0\) would clearly block the route to
Pascal’s conclusion, under the usual assumption that

And nothing in the argument implies that \(f_1 \gt f_3\). (Indeed,
this inequality is questionable, as even Pascal seems to allow.) In
short, Pascal’s wager has no pull on strict
atheists.[9]

5.3 Premise 3: Rationality Requires Maximizing Expected Utility

Finally, one could question Pascal’s decision theoretic assumption
that rationality requires one to perform the act of maximum expected
utility (when there is one). Now perhaps this is an analytic
truth, in which case we could grant it to Pascal without further
discussion—perhaps it is constitutive of rationality
to maximize expectation, as some might say. But this premise has met
serious objections. The Allais 1953 and Ellsberg 1961 paradoxes, for
example, are said to show that maximizing expectation can lead one to
perform intuitively sub-optimal actions. So too the St. Petersburg
paradox, in which it is supposedly absurd that one should be prepared
to pay any finite amount to play a game with infinite expectation.
(That paradox is particularly apposite
here.)[10]

Various refinements of expected utility theory have been suggested as
a result of such problems. For example, we might consider expected
differences between the pay-offs of options, and prefer one
option to another if and only if the expected difference of the former
relative to the latter is positive—see Hájek and Nover 2006,
Hájek 2006, Colyvan 2008, and Colyvan & Hájek 2016. Or we might consider
suitably defined utility ratios, and prefer one option to
another if and only if the utility ratio of the former relative to the
latter is greater than 1—see Bartha 2007. If we either admit
refinements of traditional expected utility theory, or are pluralistic
about our decision rules, then premise 3 is apparently false as it
stands. Nonetheless, the door is opened to some suitable reformulation
of it that might serve Pascal’s purposes. Indeed, Bartha argues that
his ratio-based reformulation answers some of the most pressing
objections to the Wager that turn on its invocation of infinite
utility.

Finally, one might distinguish between practical
rationality and theoretical rationality. One could then
concede that practical rationality requires you to maximize expected
utility, while insisting that theoretical rationality might require
something else of you—say, proportioning belief to the amount
of evidence available. This objection is especially relevant, since
Pascal admits that perhaps you “must renounce reason” in order to
follow his advice. But when these two sides of rationality pull in
opposite directions, as they apparently can here, it is not obvious
that practical rationality should take precedence. (For a discussion of
pragmatic, as opposed to theoretical, reasons for belief, see Foley
1994.)

5.3 Is the Argument Valid?

A number of authors who have been otherwise critical of the Wager
have explicitly conceded that the Wager is valid—e.g. Mackie
1982, Rescher 1985, Mougin and Sober 1994, and most emphatically,
Hacking 1972. That is, these authors agree with Pascal that wagering
for God really is rationally mandated by Pascal’s decision matrix in
tandem with positive probability for God’s existence, and the decision
theoretic account of rational action.

However, Duff 1986 and Hájek 2003 argue that the argument is
in fact invalid. Their point is that there are strategies besides
wagering for God that also have infinite expectation—namely,
mixed strategies, whereby you do not wager for or against God
outright, but rather choose which of these actions to perform on the
basis of the outcome of some chance device. Consider the mixed
strategy: “Toss a fair coin: heads, you wager for God; tails, you wager
against God”. By Pascal’s lights, with probability 1/2 your expectation
will be infinite, and with probability 1/2 it will be finite. The
expectation of the entire strategy is:

That is, the ‘coin toss’ strategy has the same expectation
as outright wagering for God. But the probability 1/2 was incidental to
the result. Any mixed strategy that gives positive and finite
probability to wagering for God will likewise have infinite
expectation: “wager for God iff a fair die lands 6”, “wager for God iff
your lottery ticket wins”, “wager for God iff a meteor quantum tunnels
its way through the side of your house”, and so on.

It can be argued that the problem is still worse than this, though, for there is a sense
in which anything that you do might be regarded as a mixed
strategy between wagering for God, and wagering against God, with
suitable probability weights given to each. Suppose that you choose to
ignore the Wager, and to go and have a hamburger instead. Still, you
may well assign positive and finite probability to your winding up
wagering for God nonetheless; and this probability multiplied by
infinity again gives infinity. So ignoring the Wager and having a
hamburger has the same expectation as outright wagering for God. Even
worse, suppose that you focus all your energy into avoiding
belief in God. Still, you may well assign positive and finite
probability to your efforts failing, with the result that you wager for
God nonetheless. In that case again, your expectation is infinite
again. So even if rationality requires you to perform the act of
maximum expected utility when there is one, here there isn’t one.
Rather, there is a many-way tie for first place, as it
were. All hell breaks loose: anything you might do is maximally
good by expected utility lights![11]

Monton 2011 defends Pascal’s Wager against this line of
objection. He argues that an atheist or agnostic has more than one
opportunity to follow a mixed strategy. Returning to the first example
of one, suppose that the fair coin lands tails. Monton’s thought is
that your expected utility now changes; it is no longer infinite, but
rather that of an atheist or agnostic who has no prospect of the
infinite reward for wagering for God. You are back to where you
started. But since it was rational for you to follow the mixed
strategy the first time, it is rational for you to follow it again
now—that is, to toss the coin again. And if it lands tails
again, it is rational for you to toss the coin again … With
probability 1, the coin will land heads eventually, and from that
point on you will wager for God. Similar reasoning applies to wagering
for God just in case an n-sided die lands 1 (say): with probability 1
the die will eventually land 1, so if you repeatedly base your mixed
strategy on the die, with probability 1 you will wind up wagering for
God after a finite number of rolls. Robertson 2012 replies that not
all such mixed strategies are (probabilistically) guaranteed to lead
to your wagering for God in the long run: not ones in which the
probability of wagering for God decreases sufficiently fast on
successive trials. Think, for example, of rolling a 4-sided die, then
a 9-sided die, and in general an \((n+1)^2\)-sided die on the
\(n\)th trial …, a strategy for which the probability that
you will eventually wager for God is only 1/2, as Robertson
shows. However, Easwaran and Monton 2012 counter-reply that with a
continuum of times at which the dice can be rolled, the sequence of
rolls that Robertson proposes can be completed in an arbitrarily short
period of time. In that case, what should you do next? By Monton’s
argument, it seems you should roll a die again. Easwaran and Monton
prove that if there are uncountably many times at which one implements
a mixed strategy with non-zero probability of wagering for God, then
with probability 1, one ends up wagering for God at one of these
times. (And they assume, as is standard, that once one wagers for God
there is no going back.) They concede that imagining uncountably rolls
of a die, say, involves an idealization that is surely not physically
realizable. But they maintain that you should act in the way that an
idealized version of yourself would eventually act, one
who can realize the rolls as described—that is, wager
for God outright.

There is a further twist on the mixed strategies objection. To
repeat, the objection’s upshot is that even granting Pascal all his
premises, still wagering for God is not rationally required. But we
have seen numerous reasons not to grant all his
premises. Very well then; let’s not. Indeed, let’s suppose that you
give tiny probability p to them all being true,
where \(p\) is positive and finite. So you assign
probability \(p\) to your decision problem being exactly as
Pascal claims it to be. But if it is, according to the mixed
strategies objection, all hell breaks loose. Yet again, \(p\)
multiplied by infinity gives infinity. Hence, it seems that each
action that gets infinite expected utility according to Pascal
similarly gets infinite expected utility according to you;
but by the previous reasoning, that is anything you might do. The full
force of the objection that hit Pascal now hits you too. There are
some subtleties that we have elided over; for example, if you also
assign positive and finite probability to a source
of negative infinite utility, then the expected utilities
instead become \(\infty\) – \(\infty\), which is undefined. But that
is just another way for all hell to break loose for you: in that case,
you cannot evaluate the choiceworthiness of your possible actions at
all. Either way, you face decision-theoretic paralysis. We might call
this Pascal’s Revenge. See Hájek (2015) for
more discussion.

5.4 Moral Objections to Wagering for God

Let us grant Pascal’s conclusion for the sake of the argument:
rationality requires you to wager for God. It still does not obviously
follow that you should wager for God. All that we have granted
is that one norm—the norm of rationality—prescribes
wagering for God. For all that has been said, some other norm
might prescribe wagering against God. And unless we can show that the
rationality norm trumps the others, we have not settled what you should
do, all things considered.

There are several arguments to the effect that morality
requires you to wager against God. Pascal himself appears to
be aware of one such argument. He admits that if you do not believe in
God, his recommended course of action “will deaden your
acuteness” (This is Trotter’s translation. Pascal’s
original French wording is “vous abêtira”, whose
literal translation is even more startling: “will make you a
beast”.) One way of putting the argument is that wagering for
God may require you to corrupt yourself, thus violating a Kantian duty
to yourself. Clifford 1877 argues that an individual’s believing
something on insufficient evidence harms society by promoting
credulity. Penelhum 1971 contends that the putative divine plan is
itself immoral, condemning as it does honest non-believers to loss of
eternal happiness, when such unbelief is in no way culpable; and that
to adopt the relevant belief is to be complicit to this immoral
plan. See Quinn 1994 for replies to these arguments. For example,
against Penelhum he argues that as long as God treats non-believers
justly, there is nothing immoral about him bestowing special favor on
believers, more perhaps than they deserve. (Note, however, that Pascal
leaves open in the Wager whether the payoff for non-believers \(is\)
just; indeed, as far as his argument goes, it may be extremely
unjust.)

Finally, Voltaire protests that there is something unseemly about the
whole Wager. He suggests that Pascal’s calculations, and his appeal to
self-interest, are unworthy of the gravity of the subject of theistic
belief. This does not so much support wagering against God, as
dismissing all talk of ‘wagerings’ altogether. Schlesinger
(1994, 84) canvasses a sharpened formulation of this objection: an
appeal to greedy, self-interested motivations is incompatible with
“the quest for piety” that is essential to religion. He
replies that the pleasure of salvation that Pascal’s Wager
countenances is “of the most exalted kind”, and that if
seeking it counts as greed at all, then it is “the manifestation
of a noble greed that is to be acclaimed” (85).

6. What Does It Mean to “Wager for God”?

Let us now grant Pascal that, all things considered (rationality and
morality included), you should wager for God. What exactly does this
involve?

A number of authors read Pascal as arguing that you should
believe in God—see e.g. Quinn 1994, and Jordan 1994a.
But perhaps one cannot simply believe in God at will; and rationality
cannot require the impossible. Pascal is well aware of this objection:
“[I] am so made that I cannot believe. What, then, would you have me
do?”, says his imaginary interlocutor. However, he contends that one
can take steps to cultivate such belief:

You would like to attain faith, and do not know the way;
you would like to cure yourself of unbelief, and ask the remedy for it.
Learn of those who have been bound like you, and who now stake all
their possessions. These are people who know the way which you would
follow, and who are cured of an ill of which you would be cured. Follow
the way by which they began; by acting as if they believed, taking the
holy water, having masses said, etc. …

But to show you that this leads you there, it is this which will
lessen the passions, which are your stumbling-blocks.

We find two main pieces of advice to the non-believer here: act like a
believer, and suppress those passions that are obstacles to becoming a
believer. And these are actions that one can perform at will.

Believing in God is presumably one way to wager for God. This
passage suggests that even the non-believer can wager for God, by
striving to become a believer. Critics may question the psychology of
belief formation that Pascal presupposes, pointing out that one could
strive to believe (perhaps by following exactly Pascal’s prescription),
yet fail. To this, a follower of Pascal might reply that the act of
genuine striving already displays a pureness of heart that God would
fully reward; or even that genuine striving in this case is itself a
form of believing.

According to Pascal, ‘wagering for God’ and
‘wagering against God’ are contradictories, as there is no
avoiding wagering one way or another: “you must wager. It is not
optional.” The decision to wager for or against God is one that
you make at a time—at \(t\), say. But of course Pascal does
not think that you would be infinitely rewarded for wagering for God
momentarily, then wagering against God thereafter; nor that you would
be infinitely rewarded for wagering for God sporadically—only on
the last Thursday of each month, for example. What Pascal intends by
‘wagering for God’ is an ongoing action—indeed, one
that continues until your death—that involves your adopting a
certain set of practices and living the kind of life that fosters
belief in God. The decision problem for you at \(t\), then, is
whether you should embark on this course of action; to fail to do so
is to wager against God at \(t\).

7. The Continuing Influence of Pascal’s Wager

Pascal’s Wager vies with Anselm’s Ontological Argument for being the most famous argument in the philosophy of religion. Indeed, the Wager arguably has greater influence nowadays than any other such argument—not just in the service of Christian apologetics, but also in its impact on various lines of thought associated with infinity, decision theory, probability, epistemology, psychology, and even moral philosophy. It has provided a case study for attempts to develop infinite decision theories. In it, Pascal countenanced the notion of infinitesimal probability long before philosophers such as Lewis 1980 and Skyrms 1980 gave it prominence. It continues to put into sharp relief the question of whether there can be pragmatic reasons for belief, and the putative difference between theoretical and practical rationality. It raises subtle issues about the extent to which one’s beliefs can be a matter of the will, and the ethics of belief.

Reasoning reminiscent of Pascal’s Wager, often with an explicit
acknowledgment of it, also informs a number of debates in moral
philosophy, both theoretical and applied. Kenny 1985 suggests that
nuclear Armageddon has negative infinite utility, and some might say
the same for the loss of even a single human life. Stich 1978
criticizes an argument that he attributes to Mazzocchi, that there
should be a total ban on recombinant DNA research, since such research
could lead to the “Andromeda scenario” of creating a
killer strain of bacterial culture against which humans are helpless;
the ban, moreover, should be enforced if the “Andromeda scenario
has even the smallest possibility of occurring” (191), in
Mazzocchi’s words. This is plausibly read, then, as an
assignment of negative infinite utility to the Andromeda
scenario. More recently, Colyvan, Cox, and Steele 2010 discuss
Pascal’s Wager-like problems for certain deontological moral
theories, in which violations of duties are assigned negative infinite
utility. Colyvan, Justus and Regan 2011 canvas difficulties associated
with assigning infinite value to the natural environment. Bartha and
DesRoches 2017 respond, with an appeal to relative utility
theory. Stone 2007 argues that a version of Pascal’s Wager
applies to sustaining patients who are in a persistent vegetative
state; see Varelius 2013 for a dissenting view.

Pascal’s Wager is a watershed in the philosophy of religion. As we have seen, it is also a great deal more besides.